We prove a formal Darboux-type theorem for Hamiltonian operators of
hydrodynamic type, which arise as dispersionless limits of the
Hamiltonian operators in the KdV and similar hierarchies. We prove
that the Schouten Lie algebra is a formal differential graded Lie
algebra, which allows us to obtain an analogue of the Darboux normal
form in this context.
¶ We include an exposition of the formal deformation theory of
differential graded Lie algebras $\mathfrak {g}$ concentrated in
degrees $[-1,\infty)$; the formal deformations of $\mathfrak {g}$ are
parametrized by a 2-groupoid that we call the Deligne 2-groupoid of
$\mathfrak {g}$, and quasi-isomorphic differential graded Lie algebras
have equivalent Deligne 2-groupoids.